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The equation to use is sin(15) = sin(45 - 30)
Using 45 and 30 works well because 45 and 30 degree angles are parts of triangles with well-defined ratios. For example, the 45 degree angle is part of a 45:45:90 triangle with ratios of 1:1:square_root(2). Similarly, the 30 degree angle features in a 30:60:90 triangle with ratios of 1:square_root(3):2.

Expand the expression using the angle difference identity

Here, sin( 45 - 30 ) = sin(45) * cos(30) + cos(45) * sin(30).

Calculate the individual elements

In geometry, sin is defined as the ratio of the side opposite the angle to the hypotenuse. Therefore, sin(45) = 1 / square_root(2) and sin(30) = 1 / 2.
Similarly, cos is defined as the ratio of the side adjacent to the angle to the hypotenuse. Hence cos(45) = 1 / square_root(2), and cos(30) = square_root(3) / 2.